General Setting. Now we are ready to introduce the setting of the deep neural networks employed in the forthcoming theoretical analysis, which is adopted from [1]}. We focus on the deep neural networks with \(N_{conn}\)  fully-connected layers and \(N_{conv}\)  convolutional layers. The \(i\) -th convolutional layer has a kernel \(\mathbf {K}^{(i)} \in \mathbb {R}^{k_i \times k_i \times c_{i-1} \times c_i}\) . Recall that convolution is a linear operator. For a Given kernel \(\mathbf {K}\) , we denote its associated matrix as \(op(\mathbf {K})\) , such that \(\mathbf {K}(\mathbf {x}) = op(\mathbf {K}) \mathbf {x}\) . Moreover, we assume that, each time, the convolution layer is followed by a componentwise non-linear activation function and an optional pooling operation. We assume that the activation functions and the pooling operations are all 1-Lipschitz. For the \(i\) -th fully-connected layer, we denote its weight as \(\mathbf {V}^{(i)}\) . Above all, the complete parameter set of a given deep neural network could be represented as \(\mathbf {P}= \lbrace \mathbf {K}^{(1)},\cdots , \mathbf {K}^{(N_{conv})}, \mathbf {V}^{(1)}, \cdots , \mathbf {V}^{(N_{conn})}\rbrace \) . Again, we also assume that the loss function is \(\phi _{\ell }\) -Lipschitz and \(Range\lbrace \ell \rbrace  \subseteq [0,B]\) . Finally, given two deep neural networks with parameters \({{\color {red}\mathbf {P}}}\)  and \({\color {c2}\tilde{\mathbf {P}}}\) , we adopt a metric \(d_{NN}(\cdot ,\cdot )\)  to measure their distance:
\(d_{NN}({{\color {red}\mathbf {P}}},{\color {c2}\tilde{\mathbf {P}}}) = \sum _{i=1}^{N_{conv}} ||op({{\color {red}\mathbf {K}^{(i)}}}) -op({\color {c2}\tilde{\mathbf {K}}^{(i)}})||_2 + \sum _{i=1}^{N_{conn}} ||{{\color {red}\mathbf {V}^{(i)}}}- {\color {c2}\tilde{\mathbf {V}}^{(i)}}||_2\) 
